Discussion
Consider the following statements :
(1) Changing basic objective function coefficient \(c_{j}\) will affect the entire 0-raw to change.
(2) Changing right-hand side of a constraint will retain the current basis to be optimal even if the constraint is negative.
(3) Changing the column of a non-basic variable \(x_{j}\) will affect, the coefficient of \(x_{j}\) in row-0 is still non-negative and the current basis is optimal.
Which of the above are correct ?
Answer: B
Statements 1 and 3 are correct.
1: the reduced costs in row 0 are \(z_{j} - c_{j} = c_{B}B^{-1}a_{j} - c_{j}\). Changing the cost of a basic variable changes \(c_{B}\), and so every entry of row 0 can change, as the statement says.
3: changing the column of a non-basic variable \(x_{j}\) affects only its own column of the tableau, so only its row-0 coefficient changes. If that coefficient stays non-negative (for a maximisation), the current basis remains optimal.
2 is false: changing a right-hand side changes the values of the basic variables, \(B^{-1}b\). If any of them becomes negative, the current basis is no longer feasible and so no longer optimal; the dual simplex method is then needed.
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